Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 297508, 8 pages
doi:10.1155/2008/297508
Research Article
On New Extensions of Hilbert’s Integral Inequality
He Leping,1 Gao Mingzhe,2 and Zhou Yu2
1
Department of Mathematics and Applied Mathematics, College of Mathematics and Computer Science,
Jishou University, Jishou Hunan 416000, China
2
Department of Mathematics and Computer Science, Normal College of Jishou University,
Jishou Hunan 416000, China
Correspondence should be addressed to Gao Mingzhe, mingzhegao@163.com
Received 11 September 2007; Revised 5 November 2007; Accepted 11 December 2007
Recommended by Feng Qi
It is shown that some new extensions of Hilbert’s integral inequality with parameter λλ > 1/2 can
be established by introducing a proper weight function. In particular, when λ 1, a refinement of
Hilbert’s integral inequality is obtained. As applications, some new extensions of Widder’s inequality and Hardy-Littlewood’s inequality are given.
Copyright q 2008 He Leping et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction and lemmas
Let fx, gx ∈ L2 0, ∞. It is well known that the inequality of the form
∞
α
fxgy
dx dy ≤ π
x y − 2α
∞
α
f 2 xdx
1/2 ∞
1/2
g 2 xdx
1.1
α
is called Hilbert’s integral inequality, where the coefficient π is the best possible.
In 1, by introducing a parameter λ λ > 1/2, the following extension of 1.1 of the
form
∞
∞
1/2 ∞
1/2
fxgy
π
1−λ 2
1−λ 2
dx
dy
≤
x
f
xdx
x
g
xdx
1.2
λ
λ
λsin π/2λ
0 x y
0
0
was established.
Recently, various improvements and extensions of 1.1 appear in a great deal of papers
see 2. The aim of this paper is to give some new improvements of 1.1 and 1.2 and then
present some important applications.
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International Journal of Mathematics and Mathematical Sciences
We now introduce some notations that will be used throughout the paper.
Let x − α ≥ 0, y − α ≥ 0, λ > 1/2, and let cx − α be an integrable function in α, ∞.
Define Ex, y 1 − cx − α cy − α such that Ex, y ≥ 0 for x, y ∈ α, ∞ × α, ∞. We
also define
∞
f 2 x
x − α 1/2
J1 Ex, ydx dy,
λ
λ y−α
α x − α y − α
1.3
∞
y − α 1/2
f 2 y
Ex, ydx dy.
J2 λ
λ x−α
α x − α y − α
Lemma 1.1. With the above-mentioned assumptions, one has
2 ∞
2 2 ∞
π
1−λ 2
2
J 1 J2 x − α f xdx −
kxf xdx
,
λsinπ/2λ
α
α
1.4
where the weight function kx is defined by
kx x − α
1−λ
λsinπ/2λ
π
∞ c x − αt t−1/2
0
1 tλ
dt − cx − α .
1.5
Proof. By using the substitution t y − α/x − α, it is easy to deduce that
∞ ∞
x − α 1/2
1
Ex, ydy f 2 xdx
J1 λ y − α
λ
α
α x − α 1 y − α/x − α
∞ ∞
1/2
1
1
1 − cx − α c x − αt dt x − α1−λ f 2 xdx
λ
t
α
0 1 t
∞
∞ c x − αt 1 1/2
π
π
dt −
cx − α x − α1−λ f 2 xdx
λsinπ/2λ
t
λsinπ/2λ
1 tλ
α
0
∞
∞
π
x − α1−λ f 2 xdx kxf 2 xdx ,
λsinπ/2λ
α
α
1.6
where kx is a function defined by 1.5.
Similarly, we have
∞
∞
π
J2 x − α1−λ f 2 xdx −
kxf 2 xdx .
λsinπ/2λ
α
α
1.7
From the above equations involving J1 and J2 , 1.4 holds true.
Lemma 1.2. Let x − α ≥ 0 and λ > 1/2. Then
⎧
x − αλ−1/2 − 1
π
⎪
⎪
⎪
∞
, x−α /
1,
⎨
t−1/2
λsinπ/2λ
x − αλ − 1
dt
⎪
π
1
⎪
1 x − αλ tλ
0 1 tλ
⎪
⎩
1−
,
x − α 1.
λsinπ/2λ
2λ
1.8
He Leping et al.
3
Proof. The case x − α /
1 was studied in 3, or can be obtained by using 4. Next, consider the
case x − α 1. By the definition and properties of beta function, it is easy to deduce that
∞
0
t−1/2
1 tλ
2 dt 1
1
1
1
π
1
1
1
1
1−
B
1−
B
,2 −
,1 −
.
λ
2λ
2λ
λ
2λ
2λ
2λ
2λ λsinπ/2λ
1.9
2. Theorem and its corollary
Theorem 2.1. Let fx and gx be two real functions such that 0 <
∞
and 0 < α x − α1−λ g 2 xdx < ∞, where λ > 1/2. Then
∞
∞
α
x − α1−λ f 2 xdx < ∞
4
fxgy
dx dy
x − αλ y − αλ
4 ∞
2 ∞
2 π
1−λ 2
2
≤
x − α f xdx −
ωλ xf xdx
λsinπ/2λ
α
α
∞
2 ∞
2 α
x − α1−λ g 2 xdx
×
−
ωλ xg 2 xdx
α
2.1
,
α
where the weight function ωλ x is defined by
⎧
λ−1/2
−1
1
⎪
1−λ x − α
⎪
⎪x − α
−
, x−α /
1,
⎨
x − αλ − 1
1 x − αλ
ωλ x ⎪
⎪
⎪
⎩1 − 1 ,
x − α 1.
2 2λ
2.2
Proof. First, assume f g. Let Fx, y fxfy/x − αλ y − αλ , Ex, y 1 − cx − α cy − α.
Then the following holds:
∞
∞
Fx, ydx dy Fx, yEx, ydx dy.
2.3
α
α
In fact, it is obvious that
∞
Fx, yEx, ydx dy
α
∞
Fx, ydx dy −
α
∞
Fx, ycx − αdx dy α
∞
2.4
Fx, ycy − αdx dy.
α
We need only to show that
∞
∞
Fx, ycx − αdx dy Fx, ycy − αdx dy.
α
α
2.5
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International Journal of Mathematics and Mathematical Sciences
Let ϕx ∞
α
∞
α
ft/x − αλ t − αλ dt. Then
Fx, ycx − αdx dy
∞ ∞
fy
dy fxcx − αdx
λ
λ
α
α x − α y − α
∞ ∞
∞
ft
dt fxcx − αdx ϕxfxcx − αdx
λ
λ
α
α x − α t − α
α
∞
∞ ∞
ft
ϕyfycy − αdy dt fycy − αdy
λ
λ
α
α
α y − α t − α
∞ ∞
∞
fx
dx
fycy
−
αdy
Fx, ycy − αdx dy.
λ
λ
α
α y − α x − α
α
2.6
Noting that Ex, y ≥ 0 and applying Schwarz’s inequality, we have
∞
2
Fx, ydx dy
∞
2
Fx, yEx, ydx dy
α∞ 1/2
fx
x − α 1/4 Ex, y
1/2 y − α
α
x − αλ y − αλ
2
1/2
fy
y − α 1/4 × Ex, y
dx dy
1/2 x − α
x − αλ y − αλ
∞ f 2 x
x − α 1/2
≤
Ex, y dx dy
x − αλ y − αλ y − α
α
∞ f 2 y
y − α 1/2
×
Ex, y dx dy J1 J2 .
x − αλ y − αλ x − α
α
2.7
α
It follows from 1.4 that
∞
2
Fx, ydx dy
α
≤
π
λsinπ/2λ
2 ∞
x − α
1−λ 2
2
f xdx
−
α
2 kxf xdx
,
∞
2
α
2.8
where the weight function kx is defined by 1.5.
Let cx 1/1 xλ , where x ≥ α and λ > 1/2. It is obvious that Ex, y ≥ 0. By
Lemma 1.2, it is easy to deduce that
kx x − α
1−λ
x − α
1−λ
λsinπ/2λ
π
λsinπ/2λ
π
∞ c x − αt t−1/2
1 tλ
0
∞
0
1 tλ
dt − cx − α
t−1/2
1 x − αλ tλ
2.9
dt − cx − α
ωλ x.
He Leping et al.
5
Substitute kx ωλ x into 2.8 to obtain
∞
2 Fx, ydx dy ≤
α
π
λsin π/2λ
2 ∞
x − α
2
1−λ 2
f xdx
−
∞
α
2 ωλ xf xdx
2
.
α
2.10
Next, consider the case f /
g. By Schwarz’s inequality, we have
∞
4
fx gy
dx dy
x − αλ y − αλ
1 ∞
∞
2 2
x−αλ −1/2
y−αλ −1/2
t
fxdx t
gydy dt
α
0
≤
0
α
1 ∞
∞
α
α
2.11
2 2 1 ∞
2 2
x−αλ −1/2
y−αλ −1/2
t
fxdx dt
t
gydy dt
α
0
α
2 ∞
fx fy
x − αλ y − α
dx dy
λ
2
gx gy
x − αλ y − α
α
dx dy
λ
.
Based on 2.10, it follows from 2.11 that the inequality 2.1 is valid at once. Theorem is
proved.
The special case λ 1 in Theorem 2.1 yields the following Hilbert’s integral inequality.
∞
∞
Corollary 2.2. If 0 < α f 2 xdx < ∞ and 0 < α g 2 xdx < ∞, then
∞
α
fxgy
dx dy
x y − 2α
4
≤ π
4
∞
2
f xdx
−
2
∞
α
×
∞
2 ω1 xf xdx
2
α
2
g xdx
−
2
∞
α
2.12
2 ω1 xg xdx
2
,
α
where the weight function ω1 x is defined by
1
1
−
ω1 x √
.
x−α 1 x−α 1
2.13
Proof. It follows directly from the proof of Theorem 2.1 and so the details are omitted.
Remark 2.3. By setting f g in Corollary 2.2, 2.12 yields
∞
α
fxfy
dx dy
x y − 2α
2
≤ π
2
∞
2
f xdx
2
α
where the weight function ω1 x is defined by 2.13.
−
∞
α
2 ω1 xf xdx
,
2
2.14
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International Journal of Mathematics and Mathematical Sciences
Remark 2.4. For the case λ 2 in Theorem 2.1, 2.1 becomes
∞
4
fxgy
dx
dy
2
2
α x − α y − α
∞
2 ∞
2 π4
−1 2
2
≤
x − α f xdx −
ω2 xf xdx
4
α
α
∞
2 ∞
2 −1 2
2
×
x − α g xdx −
ω2 xg xdx
,
α
2.15
α
where the weight function ω2 x is defined by
⎧
3/2
−1
1
⎪
−1 x − α
⎪
⎨x − α
, x − α/
1,
−
x − α2 − 1
1 x − α2
ω2 x ⎪
⎪
⎩1,
x − α 1.
4
2.16
3. Some applications
As applications, we will give some extensions and refinements of Widder’s inequality and
Hardy-Littlewood’s inequality.
∞
n
∗
n
Let an ≥ 0 n 0, 1, 2, . . . , Ax ∞
n0 an x , A x n0 an x /n!. Then
∞
1
−x ∗ 2
A2 xdx ≤ π
3.1
e A x dx.
0
0
Inequality 3.1 is called Widder’s inequality see 5.
We will give an extension of 3.1 below.
Theorem 3.1. Under the above assumptions, if fx e−x−α A∗ x − α, then
∞
1
2
2 ∞
2 2
2
2
2
A xdx ≤ π
f xdx −
ω1 xf xdx
,
0
α
3.2
α
where ω1 x is defined by 2.13.
Proof. First, observe that the following holds:
∞
∞ ∞
∞
∞
an xtn
an xn ∞ n −t
−t ∗
−t
e A txdt e
t e dt an xn Ax.
dt n!
n! 0
0
0
n0
n0
n0
3.3
Let tx s − α. Then we have
1 ∞
1 ∞
2
2
1
A2 xdx e−t A∗ txdt dx e−s−α/x A∗ s − αds
dx
2
x
0
0
0
0
α
∞ ∞
∞ ∞
2
2
−s−αy ∗
−s−αu−s−α ∗
e
A s − αds dy e
A s − αds du
1
1
α
∞ ∞
0
α
e−s−αu fsds
2
du ∞
α
0
α
fsft
ds dt,
s t − 2α
3.4
He Leping et al.
7
where fx e−x−α A∗ x − α. Using Remark 2.3, the inequality 3.2 follows from 3.4 at
once.
In particular, when α 0, we obtain a refinement of 3.1.
Corollary 3.2. With the assumptions as Theorem 3.1, if fx e−x A∗ x, then
1
2
A xdx
≤π
2
2
∞
0
2
f xdx
−
2
∞
0
2 ω1 xf xdx
2
3.5
,
0
where ω1 x is defined by 2.13.
Let fx ∈ L2 0, 1. If an inequality see 6 of the form
1
0
xn fxdx, n 0, 1, 2, . . . , then we have Hardy-Littlewood’s
∞
a2n ≤ π
1
n0
f 2 xdx,
3.6
0
where π is the best constant that keeps 3.6 valid. In 7, the inequality 3.6 was extended to
the following inequality:
∞
1
f xdx ≤ π
2
0
h2 xdx,
3.7
0
1
where fx 0 tx hxdx, x ∈ 0, ∞.
The inequality 3.7 is called Hardy-Littlewood’s integral inequality. Afterwards, the inequality 3.7 was refined into the following form see 8:
∞
f 2 xdx ≤ π
0
1
t h2 tdt.
3.8
0
We will give a new extension of 3.8 here.
Theorem 3.3. Let λ > 1/2, ht ∈ L2 0, 1, and ht / 0. Define a function by fx 1 x−αλ
∞
1−λ 2
t
|ht|dt. If 0 < α x − α f xdx < ∞, then
0
∞
4
f xdx
2
α
≤
π
λsinπ/2λ
2 ∞
x − α
2
1−λ 2
f xdx
α
−
∞
ωλ xf xdx
α
2
2 1
2
t h2 tdt ,
0
3.9
where the weight function ωλ x is defined by 2.2.
Proof. By writing f 2 x in the form
f 2 x 1
0
λ
fxtx−α htdt
3.10
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International Journal of Mathematics and Mathematical Sciences
and applying Schwarz’s inequality, we obtain
∞
4
f xdx
2
∞ 1
fxt
α
α
0
1 ∞
1 ∞
4
ht dt
1/2 dx t
x−αλ −1/2
fxt
2
2 1
2
dx dt t h tdt
α
fxfyt
x−αλ y−αλ −1
2
1
2
dx dy dt t h tdt
α
∞
α
3.11
0
1 ∞
0
α
0
x−αλ −1/2
fxt
0
≤
4
htdt dx
x−αλ 0
2 1
fxfy
x − αλ y − α
dx dy
λ
2
t h2 tdt .
0
By 2.10, the inequality 3.9 follows from 3.11 at once.
Remark 3.4. By setting λ 1 and α 0, we obtain the following refinement of 3.8:
∞
4
f xdx
2
0
≤π
2
∞
0
2
f xdx
2
−
∞
ω1 xf xdx
0
2
2 1
2
t h tdt ,
2
3.12
0
where the weight function ω1 x is defined by 2.13.
Acknowledgments
Authors would like to express their thanks to the referees for their valuable suggestions and
comments. The research is supported by the Scientific Research Fund of Hunan Provincial
Education Department, Grants no. 07C520 and 06C657.
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